第七章小波与信号处理a.doc

第七章小波与信号处理
WAVELET & SIGNAL PROCESSING WITH APPLICATIONS
小波是应用数学的一个新领域
,通过物理直观概念和信号处理的实际需要建立了经验的反演公式。但当时的数学家并未认可-1807年Fourier热力学工程师-Fourier Kingdom
1986年数学家Mayer偶然构造出一个真正的小波基函数,并与Mallat合作建立了构造小波基函数的统一方法-多尺度分析方法及Mallat金字塔算法,小波才得到真正的发展。Daubechies-比利时女数学家-小波十讲-Daubechies小波
近代科学400年来-从复杂事物中寻找简单规律-几乎忘记了非线性用局部线性化来取代非线性,用片面的美掩盖了整体的美。一叶障目不见泰山。自然世界的本性,丰富多彩、神奇多样化都来源于非线性-分形、Fractals-Choas。
1963年气象学家洛伦兹用计算机仿真大气模型,仅忽略了一些看起来微不足道的参数,但结果却大相径庭。-科学别忘记了非线性
小波经过近十年的研究,数学体系已经建立,理论基础扎实,应用广泛。小波分析,时间频率局部变换,多尺度分析,解决了传统的傅立叶变换难以解决的许多难题-数学显微镜,信号分析,数据压缩、模式识别、特征提取。
A ment on Information Representation —— A Prelude
· information ¬¾¾¾ F(u)
Today'puting devices are largely limited to process f(u) type of information.
Where f represents the information, and u is a general index at which the information is described.
· examples if index u:
time t (speech, music, etc.)
position x(still image, etc.)
(time, position)(t, x) (video, etc.)
other physical parameters ...
·index u: continuous versus discrete
·index u versus information f: which one is more important
·A character of useful information: redundancy.
The Notes were prepared based on the following references:
. walker, "Fourier analysis and wavelet analysis." Notices of AMS, vol. 44, , -670, June/July 1997.
, "plotting and scheming with wavelets." Math. Magazine, b9, .
, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996.
W.-, ELEC 639A Course Notes on Wavelets, Dept of Elec. & Comp. Engn., Univ. of Victoria, Jan 1997.
§7- Are Wavelets?
· A wavelet is a function Y(t) whose average is Zero, .,
Examples:
·Wavelets that are useful in DSP and other applications often possess additional features:
§Generate (via dilation and translation) orthonormal or biorthonormal (basis) systems in L2
§Local both in time and frequency domain
§Fast position/Reconstruction Algorithms
§Vanishing moments

§7- Repr